People have been modeling objects such as matrices and lattices since time immemorial. These objects can be modeled either two dimensionally, or three dimensionally. If the matrix or lattice is a simple one or two dimensional object, then two dimensional modeling is perfectly adequate. However, if there is a third dimensional component to the lattice or matrix, two dimensional modeling, such as drawn representation, are frequently inadequate for a full understanding of the object modeled. True three dimensional models provide the truest picture of the lattice or matrix.
While it is possible to simply manufacture an exact copy of an object under study, exact copies are prohibitively expensive and are only useful for a single purpose, representing that specific object. It is cheaper and more efficient to construct three dimensional models from reusable building components because the model may be dismantled and the parts reused to represent a different lattice or matrix. Furthermore, since these reusable components would be essentially identical to each other, the manufacturer can utilize economies of scale in mass producing modeling kits.
There are three dimensional modeling kits already in existence. They have a number of advantages and disadvantages, which will be discussed below.
One style of modular building kit, or construction toy is best known as Lego.TM.. Legos.TM. are an extremely versatile construction set based upon 90.degree. angles. The variety of objects that can be produced from this style modular building kit are seemingly endless. Each individual modules of a Legos.TM. kit is orthorhombic in shape, and fit together in a face to face fashion, where projections on a top face fit into corresponding recesses in a bottom face of a second module. Being rectangular, Lego.TM. blocks are best used to forming extended planes or shapes with 90.degree. angles. While it is possible to construct approximations of 45.degree. and other diagonals by "stair-stepping" the Lego.TM. blocks, these forms are only approximations and are jagged with many projecting 90.degree. comers. Furthermore, since the Lego.TM. blocks have a male top and a female bottom, with four neuter sides, the modules can be co-attached in only one orientation, the male top of one module to the female bottom of another module.
U.S. Pat. No. 3,570,170 also describes Lego.TM. style blocks with projections on the top face that fit into recesses in an adjoining bottom face of a rectangular block.
Another style modular building kit is embodied by Bright Builders.TM.. Bright Builders.TM. have a central vertex with six radiating spokes terminated with balls. Each terminal balls has removed segments thereby providing matching notches and projections that snugly fit together. Although many different objects, lattices or matrices may be formed with these blocks, it is difficult or impossible to create octahedral or tetrahedral constructs with these toys.
Another style of modular building kit can be seen from Theodor Jacob's Constructional Toy with Slotted Interfitting parts, U.S. Pat. No. 3,570,169. The kit of this patent has planar panels with notches and projections along the panel edges that interfit edge to edge. However, these panels appear to be designed to fit together in a plane rather than three dimensions. Also, the panels are either rectangular or seem to be triangles that fit together to make rectangles. These blocks, like most of the others, would make only the familiar rectilinear constructions.
George Adams' Inflatable Interlockable Blocks, U.S. Pat. No. 5,273,477, fit together nicely at the edges. However, if the user stacks these blocks more than a story high, three blocks would share a common edge and wouldn't be held together securely.
Other types of construction kits may be found in the popular press: Alan Holden in his book Shapes, Space, and Symmetry, Columbia University Press, and also from the graphic works of M. C. Escher.
Pictured in Holden's wonderful book are models of numerous solid geometrical figures made by gluing photographs onto cardboard and following his construction directions. Using Holden's techniques to construct many geometric models drawbacks in this kind of model making have been discovered. The building modules, constructed from cardboard and Elmer's glue are fragile. Actual construction of the modules is very time consuming and sometimes messy. The bonds of Elmer's glue will weaken over time and the cardboard grows brittle and crumbly. Therefore, several elaborate models were lost to routine wear and tear. Finally, the cardboard faces can't be reused after they've been glued together.
Although Holden notes that octahedrons and tetrahedrons can be stacked to solidly fill space, in practice this is difficult. With its base lying flat, the slopes of the tetrahedron are about 70.degree.. Anything set on these slopes immediately slides off. It would be much easier to stack these shapes if there were a mechanism for attaching the solids face to face.
M. C. Escher has done extensive exploration of plane filing tiles based on hexagons, triangles, squares, rhombi and hexagons. Numerous examples of his experimentation in this field can be found on pages 27 through 51 in the book Escher on Escher .COPYRGT.1989 by Harry N. Abrams, Inc.). Escher frequently uses interlocking tiles to construct three dimensional objects. Two examples of these interlocking tiles used to form three dimensional objects are his works Three Elements and Carved Beechwood Ball with Fish. Furthermore, Flatworms, a stunning lithograph by Escher, illustrates some of the structures that can be built with Octahedral and Tetrahedral bricks.
So far as I know, however, Escher would carve or illustrate his tiles as one unit--a print or a sculpture. I have not seen separate modular units made by Escher that could come together in different combinations to form a new structure. His three dimensional carvings formed from tiles are not stackable. The faces of the solids don't interlock with other solids' faces. Also his tiles, though based on triangles and other regular polygons, are shapes of fish and other recognizable creatures. Most of these shapes do not readily slide together or apart.
Richard Buckminster Fuller's Synergetic Building Construction, U.S. Pat. No. 2,986,241, describes a number of ways to make an extended structure with octa and tetra hedrons. One manifestation of his invention are triangular panels. These panels are held together at intersecting edge vertices by nuts and bolts. In this case, connection holes must be aligned correctly for the insertion of the bolt. Fuller also mentions the possibility of holding the panels together by applying epoxy cement along adjoining edges. The epoxy cement would make it extremely difficult to disassemble the structure and reuse it.
E. L. Zimmerman's Construction Toy, U.S. Pat. No. 2,776,521, describes triangular panels that fit together to form tetrahedrons and other geometric figures, as do James T. Ziegler's Connectable Polygonal Construction Modules, U.S. Pat. No. 4,731,041. These construction modules, however, do not allow for the construction of extended lattices or matrices. That is, only two panels may be joined along any given edge. '041 also has a mechanism for attaching polyhedral blocks face to face. Arrays of six pegs fit to corresponding other arrays of six pegs, however this method holds the faces at a distance from each other. This distance prevents actual face to face contact and does not represent a true space filling model as desired. Additionally, there are a large variety of Ziegler's blocks. While male-female connects permits full face to face contact, female blocks can't connect with other female blocks and the male to male connection creates problems mentioned earlier in this paragraph.
John Wilson's Three Dimensional Polyhedral Jigsaw Type Puzzle, U.S. Pat. No. 5,104,125, describes hexagonal and pentagonal panels that fit together to make a soccer ball like polyhedron. These panels are mitered so that when joined there are no recesses or projections along the edges. However, a connecting pin is required to link polygonal panels together at the edges, which is a disadvantage. If Wilson's panels were linked together at the edges with projections and recesses, similar to Ziegler's panels ('041 mentioned in the above paragraph), they would need to be mitered differently. The panels would need to be frustums of the soccer ball with a full dihedral angle of 140.degree.54' rather than splitting the angle into two angles 69.degree.15', and 71.degree.39' as he has done. If Wilson's panels were held together at the edges with a mechanism similar to Ziegler's but mitered at 140.degree.54 they would it would permit some play at the edges because the cross sections of Ziegler's edge projections are not rhombi. For the projections to fit into each other snugly and without overlapping it is necessary for the cross sections be rhombi. Finally, it is impossible for soccer ball shapes to be stacked in a space filling manner.
Beeren's Building Blocks With Six Face Symmetry, U.S. Pat. No. 5,098,328 is a space filing shape with male-female face to face connectors. The blocks snap together easily and can also be disassembled and reused. However they, like most blocks, follow the conventional rectilinear matrix. The octa/tetra hedral matrix has a greater strength. Also, the octa/etera hedral constructions are more interesting to look at because they are much less common than the conventional rectilinear constructions. Such blocks are far more versatile than ordinary blocks, a quantum leap.
Arthur N. Willis Polygonal Building Elements with Connectors . . . (U.S. Pat. No. 3,564,758) have a number of regular polygons including equilateral triangles. His polygonal building elements have several disadvantages, however. His constructions need separate connectors to attach triangular and other regular polygonal faces to each other's corners. These connectors are long, thin and therefore somewhat fragile. It would be nice to be able to attach the faces edge to edge (or in this case, corner to corner) without a separate connecting element. Also his invention doesn't have a provision for attaching constructed solids to each other face to face.
William J. Boo's Construction Kit Educational Aid Toy (U.S. Pat. No. 4,836,787), like Willis' invention, includes a variety of regular polygonal panels (including equilateral triangles). His polygons are held together edge to edge by hooks and pile, such as Velcro.TM..